Absorption of electromagnetic waves by the dust particles in a plasma has been studied based on a Mie-Debye scattering mode. The longitudinal field of the Debye scattering has been derived and the wave energy loss from it has been calculated. It is shown that the lower the temperature of the plasma is and the higher the density of the plasma is, the larger the absorption cross section will be due to the longitudinal scattering.For the low frequency waves the electromagnetic waves scattered in a dusty plasma are mainly in the form of Debye scattering. In this case the energy loss due to the longitudinal scattering will affect the wave propagation seriously.